The Nernst relationship and concentration effects

Define the electron count per written reaction

For a consistently written cell reaction, ΔᵣG = −zFE using reversible cell potential E. Here ΔᵣG is molar reaction Gibbs energy on that written stoichiometric basis, z is the positive dimensionless number of electrons transferred per reaction amount, and F is the Faraday constant in C mol⁻¹. Voltage is J C⁻¹, so zFE has the required J mol⁻¹ basis.

The Faraday constant is F ≈ 96485.332 C mol⁻¹ and the molar gas constant is R ≈ 8.314462618 J mol⁻¹ K⁻¹. Their SI definitions are exact while these displayed decimals are rounded. For the Zn/Cu reaction with z = 2 and supplied rounded E° = 1.10 V, ΔᵣG° ≈ −212 kJ mol⁻¹ of the written reaction. Exact constants do not make a rounded electrode potential an exact measurement.

Use activities from the net reaction

Combining ΔᵣG = ΔᵣG° + RT ln Q with the reversible potential relation gives E = E° − (RT/(zF)) ln Q. Q is dimensionless. Its species powers come from the balanced net equation, while z counts the transferred electrons; these are distinct roles.

For Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s), pure-metal activities are treated as 1 in the stated reference convention, giving Q = a_Zn²⁺/a_Cu²⁺. The supplied activities 0.0100 and 1.00 give Q = 0.0100. A concentration substitute needs normalized γ_c(c/c°) and a stated dilute approximation or formal-potential convention.

The familiar coefficient has a temperature

At T = 298.15 K, RT ln(10)/F ≈ 0.05916 V, so the base-10 form is E = E° − (0.05916 V/z)log₁₀Q at that temperature. Using z = 2, Q = 0.0100 and rounded E° = 1.10 V gives approximately 1.16 V. The original graph uses the same fixed-temperature model with horizontal coordinate log₁₀Q.

At 310 K the temperature factor alone is about 0.06151 V. A full changed-temperature voltage calculation also needs E° at the new temperature; the Nernst factor does not justify silently keeping E° fixed. The graph’s many numerical curve vertices are calculated from supplied parameters, not evidence of a high-precision real voltage measurement.

Keep equilibrium and equation scaling consistent

At the thermodynamic equilibrium relation Q = K°, the reversible full-cell E is zero for the allowed written reaction, giving E° = (RT/(zF)) ln K°. The standard potential generally is not zero. An isolated zero-voltage reading does not by itself prove equilibrium, because practical devices and measurements can have other limitations.

Doubling the net reaction doubles z and the molar Gibbs-energy change on the newly written basis, squares Q and K°, and leaves E unchanged: ln(Q²)/(2z) = ln Q/z. Reversing the complete direction inverts Q and reverses the potential sign. These are thermodynamic relations; resistance, kinetics and transport are needed for loaded operation.

Worked examples

Try it yourself

1. In ΔᵣG = −zFE, is z an arbitrary sample’s electron amount in moles?

   Show the answer and reasoning

   No.

   It is the dimensionless electron stoichiometric number on the written reaction basis.

2. What kind of quotient belongs inside the Nernst logarithm?

   Show the answer and reasoning

   A dimensionless activity quotient.

   A concentration approximation must retain its reference and activity assumptions.

3. At Q = K°, which reversible full-cell potential is zero: E or necessarily E°?

   Show the answer and reasoning

   E.

   The standard value relates to ln K° and need not vanish.

4. For the written Zn/Cu model at 298.15 K, use E° = 1.10 V, z = 2 and dimensionless Q = 100. What is E from E = E° − (0.05916 V/z) log₁₀Q?

   Show the answer and reasoning

   Approximately 1.04 V.

   Since log₁₀100 = 2, subtract (0.05916/2) × 2 V from 1.10 V. The unrounded result is 1.04084 V; Q greater than 1 lowers the forward reaction’s reversible potential.

What to carry forward

• Define z and the molar reaction-energy basis explicitly.
• Use the net equation’s dimensionless species-activity powers.
• State temperature and standard-potential conditions.
• Keep reversible thermodynamics separate from loaded operation.